In the art of measuring mass flow rates of flowing substances it is known that flowing a fluid through an oscillating flow conduit induces Coriolis forces to act on the conduit. It is also known that the magnitudes of such Coriolis forces are related to both the mass flow rate of the fluid passing through the conduit and the angular velocity at which the conduit is oscillated.
One of the major technical problems previously associated with efforts to design and make Coriolis mass flow rate instruments was the necessity either to measure accurately or control precisely the angular velocity of an oscillated flow conduit so that the mass flow rate of the fluid flowing through the flow conduits could be calculated using measurements of effects caused by Coriolis forces. Even if the angular velocity of a flow conduit could be accurately determined or controlled, precise measurement of the magnitude of effects caused by Coriolis forces was another severe technical problem. This problem arises in part because the magnitudes of generated Coriolis forces are very small in comparison to other forces such as inertia and damping, therefore resulting Coriolis force-induced effects are minute. Further, because of the small magnitude of the Coriolis forces, effects resulting from external sources such as vibrations induced, for example, by neighboring machinery or pressure surges in fluid lines, may cause erroneous determinations of mass flow rates. Such error sources as discontinuities in the flow tube may even completely mask the effects caused by generated Coriolis forces rendering a flow meter useless.
A mechanical structure and measurement technique which, among other advantages: (a) avoids the need to measure or control the magnitude of the angular velocity of a Coriolis mass flow rate instrument's oscillating flow conduit; (b) concurrently provides requisite sensitivity and accuracy for the measurement of effects caused by Coriolis forces; and, (c) minimizes susceptibility to errors resulting from external vibration sources, is taught in U.S. Pat. Nos. Re 31,450, entitled "Method and Structure for Flow Measurement" and issued Nov. 29, 1983; U.S. Pat. No. 4,422,338 entitled "Method and Apparatus for Mass Flow Measurement" and issued Dec. 27, 1983; and U.S. Pat. No. 4,491,025 entitled "Parallel Path Coriolis Mass Flow Rate Meter" and issued Jan. 1, 1985. The mechanical arrangements disclosed in these patents incorporate flow conduits having no pressure sensitive joints or sections, such as bellows or other pressure deformable portions. These flow conduits are solidly mounted in a cantilevered fashion from their inlet and outlet ports. For example, in U.S. Pat. No. 4,491,025 but not limited thereto, the flow conduits can be welded or brazed to a support, so they can be oscillated in spring-like fashion about axes which are located near the solidly mounted sections of the flow conduits. Additionally these solidly mounted flow conduits are preferably designed so they have resonant frequencies about the axes located near the mountings which are lower than the resonant frequencies about the axes relative to which Coriolis forces act. By so designing the flow conduits, a mechanical situation arises whereby, under flow conditions, the forces opposing generated Coriolis forces are essentially linear spring forces. The Coriolis forces, opposed by essentially linear spring forces, deflect the flow conduit containing flowing fluid about axes located between and essentially equidistant from the portions of the flow conduits in which Coriolis forces are generated. The magnitudes of the deflections are a function of the magnitudes of the generated Coriolis forces and the linear spring forces opposing the generated Coriolis forces.
As stated above, the flow conduits, in addition to being deflected by the Coriolis forces, are also driven to oscillate. Accordingly, under flow conditions, one portion of each flow conduit on which the Coriolis forces act will be deflected so as to move ahead, in the direction in which the flow conduit is moving, of the other portion of the flow conduit on which Coriolis forces are acting. The time or phase relationship between when the first portion of the oscillating flow conduit deflected by Coriolis forces has passed a preselected point on the path of oscillation for the flow conduit to the instant when the second portion passes a corresponding preselected point is a function of the mass flow rate of the fluid passing through the flow conduit.
A number of other Coriolis mass flow meters have been developed which are governed by similar equations of motion. Among these are specific embodiments disclosed in U.S. Pat. No. 4,127,028 (Cox et al., 1978), U.S. Pat. No. 4,559,833 (Sipin, 1985), U.S. Pat. No. 4,622,858 (Mizerak, 1986), PCT Application No. PCT/US85/01046 (Dahlin, filed 1985) and U.S. Pat. No. 4,660,421 (Dahlin, et al., 1987).
Prior art mass flow meters have been limited in their accuracy by the method of processing motion sensor signals and the relationship used in such processing. This limitation becomes important for phase angle differences above the range of 3 to 4 degrees (0.0524 to 0.0698 radians). The three Smith patents named above employ a linear relationship between the time difference of two portions of the flow conduit passing through a preselected point and mass flow rate. This time difference measurement may be made by optical sensors as specifically exemplified in U.S. Pat. No. Re 31,450, electromagnetic velocity sensors as specifically exemplified in U.S. Pat. Nos. 4,422,338 and 4,491,025, or position or acceleration sensors as also disclosed in U.S. Pat. No. 4,422,338.
A double flow conduit embodiment with sensors for making the preferred time measurements is described in U.S. Pat. No. 4,491,025. The double flow conduit embodiment described in U.S. Pat. No. 4,491,025 provides a Coriolis mass flow rate meter structure which is operated in a tuning fork-like manner as is also described in U.S. Pat. No. Re 31,450. The tuning fork operation contributes to minimizing effects of external vibration forces. Minimizing effects of external vibration forces is important because these forces can induce errors in the required time measurement.
The approach which has been taken in the prior art has been to assume that flow conduits exhibit symmetric behavior in the deformations about the Coriolis axis transverse to the oscillation axis. This is because an assumed absence of damping permits each portion of the flow conduit to respond essentially identically to the forces acting on the tube as would portions located symmetrically about the transverse axis. As one skilled in the art will recognize from the disclosures herein, a general solution to phase angle difference equations, assuming no asymmetric behavior of the flow conduit, yields the following expression for mass flow: ##EQU1## where the meaning of the variables and parameters for this and all subsequent equations is given in Table 1, herein.
The mass flow rate measurement scheme embodied in eq. (1) is identical with previously published schemes, e.g. U.S. Pat. No. Re 31,450. The correspondence can be shown by eliminating some of the additional considerations of this analysis.
Assume the phase angle difference .delta..theta. is sufficiently small that EQU sin (.delta..theta./2).congruent..delta..theta./2 (2) EQU cos (.delta..theta.).congruent.1 (3)
As shown herein, the phase angle difference and time delay .delta.t are related by EQU .delta..theta.=.omega..sub.d .delta.t (4)
where EQU .delta.t=t.sub.r -t.sub.l ( 5)
Using eqs. (2) and (3) together with eq. (4) in eq. (1) gives ##EQU2##
Thus, when the phase angle difference is small, which is typically true for phase angle differences below 3 to 4 degrees, the general solution to the phase angle difference equations reduces to a time delay mass flow measurement scheme. The simple change of variables from time to phase angle (eq. 50, herein) does not alter the physics of the mass flow measurement scheme. Taking account of the nonlinear phase angle difference relationship for phase angle differences above 3 to 4 degrees, does, however, increase the range of mass flow rate measurement accuracy.
If the viscous damping associated with the Coriolis motions of the flow tube is sufficiently small, it can be neglected. The critical damping ratio .zeta..sub.c is set to zero in eq. (6) resulting in EQU .phi..sub.c =0 (7) ##EQU3##
As before, the first term in the parentheses is a constant, as is the spring constant k.sub.c. Equation (8) states that when damping can be neglected, the mass flow rate scheme developed from the phase angle difference equations is identical to a constant times the frequency response term times the time delay.
It is assumed that the combined inertia of the flow tube, appendages, and fluid is sufficiently low that it can be neglected. As one skilled in the art will recognize, as the mass becomes smaller, the natural frequency .omega..sub.c increases. In the limit, as m goes to zero, .omega..sub.c tends to infinity, such that the frequency response term in eq. (8) approaches unity. Thus, when inertia is neglected, the mass flow rate measurement scheme based on the phase angle difference equations reduces to ##EQU4## Equation (9) is identical in form to the mass flow rate equation of U.S. Pat. No. Re 31,450. This equation can be presented as a phase angle difference equation by substituting the expression ##EQU5##
The non-linear relationship employed in the Dahlin PCT Appln. No. PCT/US85/01046 is reached by employing mutually contradictory assumptions of the existence of damping and the lack of damping to develop the underlying zero crossing equations and their solutions. In addition, asymmetric effects are ignored. This results in two equations in the solution which are reduced to a single solution by assuming zero damping. The Dahlin non-linear relationship can be reached by one skilled in the art by employing the following phase angle difference equations based on a lumped parameter model shown in FIG. 2, herein: ##EQU6##
If electronics are coupled to the outputs of velocity sensors such that the phase angle difference .delta..theta. is measured, then at a zero crossing either eq. (11) or (12) presented herein can be solved for the amplitude function H.sub.c. Well-known mathematical identities can be used to expand eq. (12), and then dividing by cos (.delta..theta./2), which never is zero for realistic .delta..theta., the following expression for H.sub.c results: ##EQU7## However, H.sub.c is related to the mass flow rate through eq. (37), presented herein. Setting eq. (13) equal to eq. (37) and solving for the mass flow rate yields ##EQU8## An alternate form for eq. (14) is obtained by setting ##EQU9## Substituting eq. (15) into eq. (14) and using well-known mathematical identities yields ##EQU10## Equation (16) is the mass flow measurement scheme given by PCT Appln. No. PCT/US85/01046. Note that the first term in parentheses is a non-dimensional constant, since it is assumed that the ratio f.sub.b /u.sub.d is constant for any particular driver design. Also note that eq. (16) shows how the measurement of mass flow rate depends on the frequency response of the flow tube.
Equation (16) is not the only mass flow measurement scheme that can be generated from the phase angle difference equations (11) and (12). Equation (16) resulted from manipulations on eq. (12) only. The same procedures applied to eq. 11) yields. ##EQU11## Note that eq. (17) is identical with eq. (16) except that the sign of the last term in the denominator has changed. This is a contradiction, and at least one of the mass flow measurement schemes embodied in these equations (16) and (17) must be incorrect. The contradiction is removed for the case of zero damping. Equation (36) indicates that .phi..sub.c is zero when .zeta..sub.c is zero. By eq. (15), the phase angle becomes .pi./2. Making this substitution in eqs. (16) and (17) makes them identical.
If the simplifications are made in eqs. (16) and (17) that were made to the general solution, eq. (1), then the result is the linearized formula of eq. (9).
The source of the error with the PCT Appln. No. PCT/US85/01046 approach is that the solution is not unique. There the mass flow equation is contradicted by a companion equation developed by identical methods. As explained above, both equations cannot be correct in the general case. For the special case of no damping in the Coriolis motions, the two equations (16) and (17) become identical, but this violates the assumption of non-zero damping used throughout PCT Appln. No. PCT/US85/01046. The new solution of the phase angle difference equations given by equation (71) disclosed herein overcomes this contradiction.
Prior mass flow measurement schemes have employed assumptions which are limited in their appropriateness to the materials of which the flow conduit is made. Both U.S. Pat. No. Re 31,450 and PCT Appln. No. PCT/US85/01046 ignore the asymmetry considerations disclosed herein, to assume that the time or phase angle difference between the motion of one portion of the flow conduit and the locus of the intersection of the conduit with a plane bisecting the conduit into two equal portions is equal to the time or phase angle difference between the motion of an opposed symmetrically-located portion of the flow conduit and the same locus. The damping of the mechanical system influences the degree of asymmetry, however. For systems having little damping, such as when the flow tubes are made from metal tubing, the amount of asymmetry is too small to measure (on the order of a few millionths of a degree). For these cases, it is appropriate to assume symmetry of the motion sensor signals with respect to the driver signal. If non-metallic tubes are used, however, it is expected that the inherent damping of such materials would create an amount of asymmetry large enough to measure, and hence of sufficient magnitude for correction in the mass flow rate determination. Suitable non-metallic materials include, but are by no means limited to, high-temperature glass, such as PYREX, manufactured by the Corning Glass Company, high-temperature ceramics, or fiber-reinforced high mechanical strength, temperature-resistant plastics. Additionally, the flowing fluids through flow conduits produce viscous damping which can be accounted for by appropriate modeling, as described herein. This modeling, which employs lumped parameters, accounts for both the damping due to the flow conduit properties and the damping due to the flow of fluids through the conduits.
Prior art mass flow devices have permitted the determination of fluid density. See, for example, U.S. Pat. Nos. Re 31,450, and 4,491,009 Which disclose such a density determination. Specific density meter circuitry embodiments are disclosed by Ruesch in U.S. patent application Nos. 916,973 and 916,780, both filed Oct. 9, 1986. These circuits can be employed with the embodiments disclosed herein to determine density as well as mass flow rate.